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Team:Imperial College London/Project Auxin Modelling
From 2011.igem.org
Module 2: Auxin Xpress
Auxin, or Indole 3-acetic acid (IAA), is a plant growth hormone which is produced by several soil bacteria. We have taken the genes encoding the IAA-producing pathway from Pseudomonas savastanoi and expressed them in Escherichia coli. Following chemotaxis towards the roots and uptake by the Phyto Route module, IAA expression will promote root growth with the aim of improving soil stability.
Modelling
I. INTRODUCTION
Diverse bacterial species possess the ability to produce the auxin(indole-3-acetic acid, IAA, phyto-hormone). Different biosynthesis pathways have been identified and redundancy for IAA biosynthesis is widespread among plant-associated bacteria. A high degree of similarity between IAA biosynthesis pathways in plants and bacteria was observed. [1]
The identification of intermediates led to the identification of five different pathways using tryptophan as a precursor for IAA. The synthesis pathway we used in our project consists of two steps with an intermediate of IAM(indole-3-acetamide). [2]
--tryptophan-IAM(IaaM gene-tryptophan-2-monooxygenase)
--IAM-IAA (IaaH gene-IAM hydrolase)
With the study on the auxin production process, we can predict the response of the plant (Arabidopsis) root system to the introduced auxin by generating a visualisation program of root systems.
II. OBJECTIVES
1. In order to choose the appropriate RNA promoter with the optimal strength, the auxin production amount is modelled base on the pathway via the intermediate IAM from the precusor tryptophan.
The result of modelling answers the question: " How much auxin can be produced by the genetically modified bacteria with a typical RNA promoter strength? "
2. A graphic program is created to demonstrate the growing process of the Arabidopsis root system, based on the principles of Lindenmayer system and plant physiology.
III. DESCRIPTION
1. Auxin Synthesis Pathway
The genetic auxin pathway involves two genes, iaaM and iaaH, both of them are constitutively expressed. The iaaM gene encodes a tryptophan-2-monooxygenase (T-2-monase) that catalyzes the conversion of tryptophan (Trp) to indole-3-acetamide (IAM), which is then hydrolyzed to release indole-3-acetic acid (IAA) by the hydrolase iaaH[3], at the same time the synthesized IAM and IAA will competitively inhibit the enzyme activity of tryptophan-2-monooxygenase, therefore negatively control the expression of IAA. The enzymatic reactions involved in the pathway are illustrated in below. The constitutive gene expression of iaaM and iaaH and the enzymatic reactions can be mathematically modelled using a series of ODEs described in Equation 1, all the parameters are defined in parameter section.
In addition, based on research carried out by University of Colorado[4], the tryptophan is also negatively controlled inside of bacteria; therefore the tryptophan synthesis pathway should be integrated into above model. Furthermore, in order to reduce the numbers of parameters as most of the parameters in equation 1 are not available, the auxin pathway is then simplified to two Michaelis Menten equations, which are then combined with tryptophan pathway. The whole tryptophan auxin pathway model was described in Equation 2,[5].
The assumptions associated with this model are listed below.
(1) We neglect the short transient needed by Trp-T-2-monase (substrate-enzyme(ES) complex), IAM-iaaH(ES complex), IAM- T-2-monase (inhibitor-enzyme(EI) complex) and IAA- T-2-monase (inhibitor-enzyme(EI) complex) and thus to consider these species reach their equilibrium almost instantaneously
(2) The degradation rate of IAA is extremely slow (7 days according to the experiment) compare to bacteria growth rate, therefore we use bacteria growth rate as IAA degradation rate in the model
(3) From early stage of our modeling of auxin pethway, it is believed that the rate determining species for IAA synthesis is IAM not enzyme iaaH since the production of IAM is inhibited by itself and IAA.
Furthermore, in order to reduce the numbers of parameters as most of the them in equation 1 are not available, the auxin pathway is then simplified to two Michaelis Menten equations, which are then combined with tryptophan pathway. The whole tryptophan auxin pathway model was described in Equation 3.
2. Auxin uptake and root growth
2.1. Initial a root system
To visualise our modelling result, a root system is demonstrated to show the root growth phenomena(primary root length, branching, root density, etc) in different environmental conditions(external and internal auxin concentration).
2.2. Root order:-
Root order describes the branching “generation” of a root system, a root without branching is defined as a zero-order root
A root system starts with a single root tip of a zero-order root. Then the root grows away from the plant stem in a conical way.[10]
Fig.1 a conical approximation of the root system
2.3. Tropisms:-
Root growth depends on the environmental factors, such as gravitation, soil heterogeneities, etc. Therefore, two more variables are defined to describe the plant adaptation:[10]
α:-
how strong the roots direction changes per 1cm growth ?
larger value indicates a more deflected root and a more twisted root system
N:-
the number of trials for the roots to find the optimal angles α and β for the rotation
for the downward movement
N can be any real number, if N = 1.5, if means that N can be either 1 or 2.
Fig.2 the difference of the root systems with different values of N and σ
2.4. Lindenmayer system and root growth modeling
An L-system is a parallel rewriting system, namely a variant of a formal grammar, most famously used to model the growth processes of plant development, but also able to model the morphology of a variety of organisms, due to the two main properties: recursive nature and self-similarity.[11]
Plant models and natural-looking organic forms are easy to define, as by increasing the recursion level the form slowly 'grows' and becomes more complex.
L-systems are now commonly known as parametric L systems, defined as a tuple.
A production consists of two strings, the predecessor and the successor.
For any symbol A in V which does not appear on the left hand side of a production in P,the identity production A → A is assumed. These symbols are called constants or terminals.
An L-system is context-free if each production rule refers only to an individual symbol and not to its neighbors. Context-free L-systems are thus specified by either a prefix grammar, or a regular grammar.
Using L-systems for generating graphical images requires that the symbols in the model refer to elements of a drawing on the computer screen. It interprets each constant in an L-system model as a turtle command.
2.5. Root Growth
The modelling of auxin uptake will give prediction of the root system development in the following ways:-
"What is the primary root growth rate?"
"What does the root system look like after a certain period of time?"
"How does arabidopsis respond to different auxin concentration?"
... ...
We wrote a program using Matlab to draw the 3D root system based on the principles of Lindenmayer system(turtle commands) and the root growth modelling toolbox developed by Daniel Leitner et al from BOKU(Universität für Bodenkultur Wien, University of Natural Resources and Life Sciences, Vienna).[12]
2.6. Data fitting
Apart from the growth rate parameters we took from the literature, we analyzed the raw data from wetlab to give more accurate and suitable prameters for our own project.
When the arabidopsis samples are planted, we record the root length and number of branches every three day from day 0 to day 9. Then, root length, daily root growth rate and number of branches are plotted against time and auxin concentration.
IV. RESULTS
1. Auxin Synthesis Pathway
The graph below represents the standard output of our model. It shows how the concentration of each of the species varies with time. It showing the simulation of the enzymatic reaction for each of the species, with initial concentration of OF = 1.54×10-4 µM, MF = 3.78×10-4 µM , E = 0.378µM , Trp = 4.1µM and all others = 0 [4].
The graph (Fig.3(d)) shows IAA concentration is 72.35µM, therefore each bacteria produces 7.24×10-14µm at steady state with bacteria volume equals to 10-15 dm3 [6].
From wet lab experiment we know that the optimal concentration of IAA to promote root growth 0.1nM, and the volume of arabidopsis seed with seed coating approximately equals to 4.2×10-9 m3,[7]. Therefore the number of bacteria required to maximally fasten root growth of bacteria is 5, the value varies due to the variation of seed size of different of plant.
fig 3abcd here
2. Auxin uptake and root growth
The values from literature gives the relationship between external auxin concentration and elongation of the roots is 5*10-5 mol/L → 200 µm elongation in 30 mins. The modelling parameter of growth speed is therefore 9.6*10-3 m/day.[9]
By observing the real roots grow from the plant, the demonstration is modified to give a more reliable and accurate prediction of the root growth. Arabidopsis has a primary root with zeroth order and it is thicker than the branches. Arabidopsis normally grows to the depth of 20~30cm inside the soil and branches once only. The 3D picture shown below predicts the root growth with different elongation rate(with auxin = 0.96cm/day; without auxin = 0.46cm/day[10]). They can be compared with the photo of real root system.
Fig.4 visalisation of the arabidopsis root system
The root has a growth rate of 0.96cm/day with the external auxin concentration 5x10-5mol/L, however, this data is selected from literature. To get an accurate growth rate which is particularly fitting our project, we decided to do data fitting analysis to the arabidopsis we plant.
The data fitting plots are analysed to give an approximation of the relationship between auxin concentration and root growth. The following graph gives an example of root length against time. [8]
Fig.5 root growth speed decays against time
Fig.6 primary root length(mm) VS time(day) and external auxin concentration(mol/L)
Fig.7 primary root growth rate(mm/day) VS root growth time(day)
Fig.8 number of lateral branch VS external auxin (log)concentration
From Fig.6 and Fig.8, the following conclusion can be make:-
the optimal concentration for primary root growth = 1pM, at this concentration,
the arabidopsis root reached the maximal depth into soil
the optimal concentration for lateral root branching = 1uM-10nM,
at this concentration, the arabidopsis root gained the most lateral branches
We used the data fitting toolbox of Matlab to obtain Fig.7 primary root growth rate(mm/day) VS root growth time(day), the relationship between the growth rate and the auxin concentration can be approximated by Gaussian equation. The abnormality of the 0.1nM curve is due to the two contaminated samples which stopped growing at 7mm after Day 5. Fig.7 is consistent with the prediction of the decay of the root growth speed given by Fig.7.
